- Understand the concept of parametric equations and their applications.
- Convert between parametric and Cartesian equations.
- Learn the fundamentals of polar coordinates.
- Convert between polar and Cartesian coordinates.
- Perform calculus operations on parametric and polar equations.
Definition of Parametric Equations
A curve in the plane can be represented by a set of **parametric equations**:
\[ x = f(t), \quad y = g(t), \quad t \text{ is the parameter} \]For example, the parametric equations:
\[ x = \cos t, \quad y = \sin t, \quad 0 \leq t \leq 2\pi \]represent a unit circle.
Eliminating the Parameter
To convert from parametric to Cartesian form, solve for \( t \) in terms of \( x \) or \( y \).
Example: Given \( x = 3t \) and \( y = 2t + 1 \), eliminate \( t \):
\[ t = \frac{x}{3} \] \[ y = 2 \left(\frac{x}{3}\right) + 1 = \frac{2x}{3} + 1 \]Introduction to Polar Coordinates
In **polar coordinates**, a point is represented as \( (r, \theta) \), where:
- \( r \) is the distance from the origin.
- \( \theta \) is the angle from the positive x-axis.
For example, \( (2, \frac{\pi}{4}) \) means the point is 2 units away at an angle of \( 45^\circ \).
Conversion Between Polar and Cartesian Coordinates
To convert:
- From Cartesian to Polar: \[ r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1} \left(\frac{y}{x}\right) \]
- From Polar to Cartesian: \[ x = r\cos\theta, \quad y = r\sin\theta \]
Calculus with Parametric and Polar Equations
For parametric equations \( x = f(t) \), \( y = g(t) \), the derivative is:
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]For polar equations \( r = f(\theta) \), the derivative is:
\[ \frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin\theta + r \cos\theta}{\frac{dr}{d\theta} \cos\theta - r \sin\theta} \]Examples
Example 1: Convert \( r = 2\cos\theta \) to Cartesian form.
\[ x = 2\cos\theta \cos\theta = 2\cos^2\theta \] \[ y = 2\cos\theta \sin\theta \]Using \( \cos^2\theta = \frac{1 + \cos 2\theta}{2} \), we get:
\[ x^2 + y^2 = 2x \]Exercises
- Question 1: Convert the parametric equations \( x = 4t, y = t^2 \) into Cartesian form.
- Question 2: Find the derivative \( \frac{dy}{dx} \) for the parametric equations \( x = t^2, y = t^3 \).
- Question 3: Convert the polar equation \( r = 3\sin\theta \) to Cartesian form.
- Answer 1: \( y = \frac{x^2}{16} \).
- Answer 2: \( \frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2} \).
- Answer 3: \( x^2 + y^2 = 3y \).